ON LUKASIEWICZ'S -fc-modal SYSTEM TIMOTHY SMILEY

Transcription

1 149 ON LUKASIEWICZ'S -fc-modal SYSTEM TIMOTHY SMILEY I want to provide a proof of Lukasiewicz's assertion that his Λl-modal system is characterised by a four-valued matrix. The need for such a proof was pointed out to me by Dr. R. Harrop. The theorems of the < -modal system are the formulae that can be derived from the axioms 1. CδpCSNpδq, 2. CpΔp, by use of the rules of substitution (for propositional and functorial variables) and detachment. The matrix in question (call it M) is the cross-product of a pair of twovalued matrices: C I t / N I A C I t / N I A M ; : * t t f f t M 2 : * t t f f t f t t t f f t t t t If for convenience we number the elements of the product, writing < t, t > = 2, < t, f > = 2, < /, t > = 3, < /, / > = 4, we can then describe M as follows: C I I N I A * It is easy to show that every theorem is verified by (never takes an undesignated value in) M. We want to prove the converse, that every formula verified by M is a theorem. We first observe that the truth-table of A in the matrix M^ is such that a formula A a always takes the same truthvalue as a itself. If then β is any formula, and β^ is got from it by replacing Received March 27, 1961

2 150 TIMOTHY SMILEY every part of the form Δ oe by plain a, the formulae β and β will take identical values in M r In particular, if β is verified by Mp so is β v In the same way, if β 2 is got from β by replacing each part of the form Δoe by Cct a, then if β is verified by tλ 2 so is β 2 But by the way M is constructed β is verified by M if and only if it is verified by both M^ and M 2 Hence we have: Lemma 2: If β is verified by M then β i is verified by M (* = 1,2). Both the M z are got by adding a table for Δ to the ordinary two-valued C-N matrix, but since Δ no longer occurs in either of the formulae β. its table will not enter into their evaluation. (This remains true even when the β i contain functorial variables, for the functional completeness of the C-N matrix means that it alone already contains all possible functions which the functorial variables might take as values.) Hence if β i is verified by M z it is verified by the C-N matrix alone. But all such formulae can be proved from Lukasiewicz's axiom 2. Hence we have: Lemma 2: If β i is verified by M z then J- β v (z=2,2). At this point we need to establish some theorems in the J^-modal system. References here to Lukasiewicz are to the chain of theorems at the end of his paper, while "PC" indicates that the formula is a tautology and hence provable via-lukasiewicz's theorems 10,20,22. 1 j- CEpqCδpδq -Lukasiewicz, theorem f CEpΔpCδpδΔp From 1 by substitution. 3 [ CECppΔpCδCppδΔp From 1 by substitution. 4 \- CCpΔpAEpΔpECppΔp PC. 5 [ CpΔp Axiom 1. 6 [ AEp ΔpECpp Δp From 4, 5 by detachment CδpCδCppδΔp From 2, 3, 6 by PC. 3 8 [ CCpqCΔpΔq Lukasiewicz, theorem CCNCqqdCΔNCqqΔa From 8 by substitution. 10 \- CNCqqa PC 11 (- CΔNCqqΔct From 9, 10 by detachment. 12 \- CctΔa From axiom 1 by substitution. 13 [ CAaΔNCqqΔa From 11, 12 by PC. 14 (- CNΔqCΔpp Lukasiewicz, theorem f CNΔNCqqCΔa a From 14 by substitution. 16 [ CΔaAaΔNCqq From 15 by PC. 17 \ EΔaAaΔNCqq From 13, 16 by PC. 28 [ CEpqEδpδq From 1 by PC. 19 [ EδΔaδAaΔNCqq From 17 by substitution in 18 and detachment. 20 \- EaAaNCqq PC. 21 [ EδσδAaNCqq From 20 by substitution in 28 and detachment.

3 ON LUKASIEWICZ'S JL-MODAL SYSTEM \ ECaaAaCNCqqNCqq PC. 23 \- EδCotaδAotCNCqqNCqq From 22 by substitution in 18 and detachment. Let the formula y be got from β by replacing every part of the form Δ & by AaΔNCqq. By successive substitution in 19 above we obtain a chain of provable equivalences (one for each step in the chain of replacements by which y is got from β) leading from β to y. Consequently by PC we have - E βγ. Now let y- and γ 2 be got from y just as β^ and β 2 were got from β. In view of the way in which y itself is got from /3, this means that y^, for example, comes from β by replacing each part Δ α by Act NCqq, whereas to get j8j we replace Δoe by plain α. Hence by successive substitutions in 21 above we get - Eβ^γ^ Similarly γ 2 comes from β by replacing each part Δoe by AaCNCqqNCqq, whereas to get β 2 we replace Δα by Cαoe. Hence by successive substitution in 23 above we get - E β 2 γ 2. The point about the construction of y is that in it Δ always occurs prefixed to the same formula, viz. NCqq. Hence the implication Cγ Cγ 2 γ is as Cβ 1 Cβ 2 β in general is not a straightforward substitution instance of CδpCδCpp 8Δp, and so by substitution in 7 above we have - Cy^ Cγ 2 γ. From this and the equivalences just established we get by PC - Cβ 1 Cβ 2 β, and thus have: Lemma 3: If \- β (i = 2,2), then - β. The desired result follows as soon as the three lemmas are put together, but it may be worth illustrating the various constructions by an example. Let β be -Lukasiewicz's theorem 92, ΔCΔpp. Then β t = Cpp; β 2 = CCCpppCCppp; γ = ACAp ΔNCqqp ΔNCqq; y χ = ACApNCqqpNCqq; γ 2 = ACApCNCqqNCqqpCNCqqNCqq. The tautology yq^ is -Lukasiewicz's theorem 13 and β 2 is a substitution instance of it. To prove Eβγ we need two applications of 19, taking δ to be Op and' in order to get EΔCΔppΔCApΔNCqqp and EΔCAp ΔNCqqp ACAp ΔNCqqp ΔNCqq respectively; and similarly for the (tautologous) equivalences Eβ^y^ Finally Cγ^Cγ 2 γ comes from 7 by substituting NCqq for p and ACAp'p* for δ. If we compare this proof with Lukasiewicz's own treatment we find that he begins by showing that his axioms and rules are verified by M and then goes on to say (p. 127), *It follows from this consideration that all the formulae of our modal logic based on the axioms 1-4 are verified by the matrix M9. It also follows that no other formulae besides can be verified by M9; this results from the fact that the classical C-N propositional calculus, to which all the formulae of our modal logic are matrically reducible, is 'saturated', i.e., any formula must be either asserted on the grounds of its asserted axioms, or rejected on the ground of the axiom of rejection - p, which easily follows by substitution from our axiom 3 or 4. M9, therefore, is an adequate matrix of the-t-modal logic." Although this is the only place in his paper at which the phrase *matrically reducible" occurs, I take it that-lukasiewicz had in mind something like my own reduction of β to β t and β 2, but carried a stage further by the

4 152 TIMOTHY SMILEY elimination not only of Δ but of functorial variables as well (so that a formula with n functorial variables would have not two but 2Λ6 n formulae associated with it). It is possible too that the reference to the "saturatedness" of the C-N calculus is intended to link the matrix verification of the various associated formulae with their provability in the system, though it is not an immediately obvious reference to make, since saturatedness is not a matrix property of a calculus at all. On the other hand (and quite crucially) there is nothing at all in -Lukasiewicz's account that would link the provability of a formula with the provability of the associated C-N formulae. There is a further way, not touched on by -Lukasiewicz, in which the matrix M is characteristic for his system. As well as providing for the assertion of formulae the Ji-modal system also, allows for their rejection, by means of two rejected 'axioms': 3. CΔpp, 4. Δp, together with counterparts of the rules of detachment and substitution. We shall show that a formula is rejected in the system if and only if it is falsified by (sometimes takes an undesignated value in) M. It is easy to see that every rejected formula is falsified by M. To prove the converse we first need to carry out some rejections in the system: 24 [ CNΔNCqqCΔpp From 14 by substitution. 25 -) CΔpp Axiom N ΔNCqq From 24, 25 by detachment. 27 f- C ΔNCqq Δp From 11 by substitution. 28 ] Δp Axiom j ΔNCqq From 27, 28 by detachment. 30 [ CNCqqΔp PC 31 \ NCqq From 28, 30 by detachment. Let φ 1 = Cqq, φ 2 = N ΔNCqq, φ^ = ΔNCqq, φ^ = NCqq. We see that each formula φ i takes exclusively the value i in the matrix, and we see from 26, 29, and 31 above that if i is undesignated φ i is rejected. We recall too that the matrix functions eligible as values for the functorial variables are exactly those singulary functions that can be built up by composition out of the identity function, the four constant functions (the functions whose values are identically 1, 2, 3, or 4), and the functions C, N, and Δ. If/ is such a function let φ, be the expression got by carrying out the corresponding composition on ', O\ NΔNC\ ΔNC\ NC\ and the functors C, N, and Δ. Suppose now that β is a formula which is falsified by M; i.e., suppose that there is a 'falsifying* assignment of values to the propositional and functorial variables of β under which it receives an undesignated value. Let γ be got from β by substituting φ i for each propositional variable which is given the value i in the falsifying assignment, and by substituting φ, for

5 ON -LUKASIEWICZ'S *.-MODAL SYSTEM 153 each functorial variable which is given the function / as value in the falsifying assignment, γ has no functorial variables and only the one propositional variable, and its construction is designed so that it always takes the same value, namely the particular value taken by β in the falsifying assignment. Let this value be z, (z = 2, 3, or 4). Then both the antecedent and consequent of Cγφ. always take the value z, so that the implication itself is verified by M and thus, by our earlier result, is a theorem. But if (- Cγφ. and - φ i then by the rejection-rule of detachment we have - y, and from this since γ is a substitution-instance of /3 by the rejection-rule of substitution we get - /3, as was to be shown. Since we have shown that every formula verified by M is asserted in the -t-modal system, and that every formula falsified by M is rejected, and since every formula is either verified or falsified, it follows as a corollary that every formula is either asserted or rejected i.e. that the4l-modal system is * saturated". NOTES 1. Jan Lukasiewicz, "A system of modal logic," The Journal of Computing Systems vol. 1 no. 3 (1953), pp Lukasiewicz, op. cit. p. 123, referring to an unpublished paper by C. A. Meredith,. This paper is still unpublished, but a sufficiently close result is to be found in Meredith's "On an extended system of the propositional calculus," Proceedings of the Royal Irish Academy, vol. 54 section A no. 3(1951), pp This short proof of 7 is due to Professor A. N. Prior. 4. M9 is what I have called M, though ^Lukasiewicz constructs it in a rather different way, forming the cross-product of the C-N matrix with itself and only afterwards inserting a table for Δ as part of the 'multiplication' process. Clare College University of Cambridge Cambridge, England